uniform_polychora

The figure to the right is a cross section of the polychoron "sirgax" - also called the small rhombated grand hexacosichoron, one of the 8190 known uniform polychora. This section is a dodecahedral symmetric section 39 percent of the way down from the top. Sirgax has a wedge shaped vertex figure, and contains the following cells: 720 pentagram prisms (stips), 600 cuboctahedra (coes), and 120 great icosidodecahedra (gids). Presently I have been using Pov-Ray for rendering sections of over 1000 polychora, and plan to eventually render sections of all 8190, however this'll take some time. I'm now working on producing a CD which will contain these sections and hope to have it available by spring of 2003.

Check out my new online store - Polychoron Graphix

A polychoron is a closed four dimensional figure bounded by cells, with the following criteria:

1. Each face must join exactly two cells.

2. Adjacent cells are not corealmic (in the same realm (3-D version of plane)).

3. The figure is not a compound of figures with the above 2 criteria.

The cells are not limited to being polyhedra, but are also allowed to be exopolyhedra - a closed non-polyhedron 3-D figure having an even number of faces meeting at each edge, where adjacent faces are not coplanar. Polyhedra have only 2 faces meeting at each edge. Polychora that have any exopolyhedra as cells are called exotic polychora. The 8190 uniform polychora include the exotics. For a polychoron to be uniform, it must be vertex-transitive (vertices are equivalent), and its cells must be uniform. For a polyhedron or exopolyhedron to be uniform, it must be vertex transitive, and have regular polygons as faces. A regiment is a group of polytopes with the same set of vertices and edges.

I have been studying polychora, as well as higher dimensional polytopes, since 1990. Back in the early 90's, I was primarily searching for them and making many discoveries, drawing the vertex figures as well as listing the cells. Due to the long polyhedron names (which were the cells of the polychora), I would abbreviated their names (i.e. quasitruncated small stellated dodecahedron = QTSSD), this eventually led me to create strange sounding "short names" for these figures. (QTSSD = quit sissid). Presently I have given all of the uniform polyhedra and uniform polychora (except idcossids and dircospids) short names. Nowadays, I'm working with cross sections of these figures.

I have grouped the uniform polychora into 29 categories. Click on existing links for more details. Click here for a more printer friendly category list. The categories are as follows:

1. Regulars - this includes the 16 regular polychora and 3 facetings that didn't fit anywhere else. (19 total).

2. Truncates - this includes truncates, quasitruncates, and other truncates (21 total).

3. Triangular Rectates - this includes 7 regiments of 3 polychora - which have vertex figures resembling the triangular prism and it's facetings. (21 total).

4. Ico Regiment - this includes the 13 polychora that share the same vertices and edges as the 24-cell (ico). (13 total).

5. Pentagonal Rectates - this includes 4 regiments of 22 polychora which have vertex figures shaped like the pentagonal prism and its facetings. (88 total).

6. Sphenoverts - this includes the polychora with wedge shaped vertex figures and their facetings (sirgax is one). (165 total).

7. Bitruncates - this includes polychora with disphenoid vertex figures, (11 total).

8. Great Rhombates - this includes those with vertex figures shaped like tetrahedra with bilateral symmetry. (23 total).

9. Maximized - this includes the polychora with irregular tetrahedra vertex figures, these have the maximum number of vertices for their symmetry groups. (22 total).

10. Prismatorhombates - this includes those with trapezoid pyramid vertex figures and their facetings. Contains 30 regiments of 3, (90 total).

11. Triangular Antipodiumverts - this includes those with triangular antipodium (looks like antiprism) vertex figures and their facetings. (40 total).

12. Frustrumverts - this includes those with triangular frustum vertex figures and their facetings. (30 total).

13. Spic and Giddic Regiments - Spic and Giddic both have square antiprism shaped vertex figures, each regiment contains 22 polychora. (44 total).

14. Skewverts - these are 4 regiments of 15, which have skewed wedge (and facetings) shaped vertex figures. (60 total).

15. Afdec Regiment - this regiment has icoic (24-cell) symmetry, and contains 54 polychora.

16. Affixthi Regiment - this regiment has hyic (120-cell) symmetry, and contains 100 polychora.

17. Sishi Regiment - Sishi is the small stellated 120-cell, its regiment contains 2 regulars, 8 swirlprisms, and 34 others - the "others" are mentioned here. (34 total).

18. Ditetrahedrals - this group contains those with vertex figures shaped like the truncated tetrahedron, and the "small rhombitetratetrahedron" (looks like cuboctahedron), and their facetings. (213 total).

19. Prisms - this contains the prisms of the uniform polyhedra, I also call them "Hedrisms". (74 total).

20. Miscellaneous - this contains the 6 antiprisms (includes grand antiprism), the 6 normal snubs, the 10 swirlprisms, and 2 recent discoveries with 4,4 duoprism symmetry. (24 total).

21. Padohi Super Regiment - padohi has a pentagonal antipodium vertex figure, its regiment contains a whopping 354 polychora.

22. Gidipthi Super Regiment - gidipthi has a pentagonal frustrum vertex figure, its regiment also contains 354 polychora. Many of these are very intricate.

23. Rissidtixhi Super Regiment - rissidtixhi has a ditrigon prism vertex figure, its regiment contains 316 polychora.

24. Stut Phiddix Super Regiment - stut phiddix has a triagular cupola shaped vertex figure and its regiment contains 238 polychora.

25. Getit Xethi Super Regiment - getit xethi has a vertex figure which resembles stut phiddix's but sort of squashed a bit, this regiment has 238 polychora.

26. Blends - this contains a non-Wythoffian regiment of 16 polychora.

27. Baby Monster Snubs - this contains 2 regiments of 15 non-Wythoffian snubs, many are chiral.

28. Idcossids - this is one of the grandest regiments of all, containing a whopping 2749 non-Wythoffian snubs - these are horrificly complex, nearly all are chiral. Based on the compound of 10 padohis.

29. Dircospids - this is another horrific regiment of 2749 non-Wythoffian snubs, nearly all are chiral. This regiment most likely contains the most complex polychoron. Based on the compound of 10 gidipthis.

Check out my new polytwister page for info on some strange swirling 4-D figures!

Click here to see George Olshevsky's polychoron web site. He also has a new Multidimensional Glossary page there.

Page wrote by HedronDude, a.k.a. Jonathan Bowers

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